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using Math3.util;
using System;
using System.Collections.Generic;
using System.Numerics;

namespace Math3.primes
{
    /// <summary>
    /// Utility methods to work on primes within the <code>int</code> range.
    /// </summary>
    public class SmallPrimes
    {

        /// <summary>
        /// The first 512 prime numbers.
        /// <para>
        /// It contains all primes smaller or equal to the cubic square of Int32.MaxValue.
        /// As a result, <c>int</c> numbers which are not reduced by those primes are guaranteed
        /// to be either prime or semi prime.
        /// </summary>
        public static readonly int[] PRIMES = 
        {
            2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
            79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163,
            167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251,
            257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349,
            353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443,
            449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 
            563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647,
            653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757,
            761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 
            877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 
            991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 
            1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181,
            1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283,
            1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 
            1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487,
            1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583,
            1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 
            1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789,
            1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 
            1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 
            2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 
            2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 
            2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 
            2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 
            2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579,
            2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 
            2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 
            2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 
            2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 
            3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 
            3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257,
            3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 
            3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491,
            3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 
            3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671
        };

        /// <summary>
        /// The last number in PRIMES.
        /// </summary>
        public static readonly int PRIMES_LAST = PRIMES[PRIMES.Length - 1];

        /// <summary>
        /// Hide utility class.
        /// </summary>
        private SmallPrimes() { }

        /// <summary>
        /// Extract small factors.
        /// </summary>
        /// <param name="n">the number to factor, must be &gt; 0.</param>
        /// <param name="factors">the list where to add the factors.</param>
        /// <returns>the part of n which remains to be factored, it is either a prime or a semi-prime</returns>
        public static int smallTrialDivision(int n, List<Int32> factors)
        {
            foreach (int p in PRIMES)
            {
                while (0 == n % p)
                {
                    n /= p;
                    factors.Add(p);
                }
            }
            return n;
        }

        /// <summary>
        /// Extract factors in the range <c>PRIME_LAST+2</c> to <code>maxFactors</code>. 
        /// </summary>
        /// <param name="n">the number to factorize, must be >= PRIME_LAST+2 and must
        /// not contain any factor below PRIME_LAST+2</param>
        /// <param name="maxFactor">the upper bound of trial division: if it is reached,
        /// the method gives up and returns n.</param>
        /// <param name="factors">the list where to add the factors.</param>
        /// <returns>n or 1 if factorization is completed.</returns>
        public static int boundedTrialDivision(int n, int maxFactor, List<Int32> factors)
        {
            int f = PRIMES_LAST + 2;
            // no check is done about n >= f
            while (f <= maxFactor)
            {
                if (0 == n % f)
                {
                    n /= f;
                    factors.Add(f);
                    break;
                }
                f += 4;
                if (0 == n % f)
                {
                    n /= f;
                    factors.Add(f);
                    break;
                }
                f += 2;
            }
            if (n != 1)
            {
                factors.Add(n);
            }
            return n;
        }

        /// <summary>
        /// Factorization by trial division.
        /// </summary>
        /// <param name="n">the number to factor</param>
        /// <returns>the list of prime factors of n</returns>
        public static List<Int32> trialDivision(int n)
        {
            List<Int32> factors = new List<Int32>(32);
            n = smallTrialDivision(n, factors);
            if (1 == n)
            {
                return factors;
            }
            // here we are sure that n is either a prime or a semi prime
            int bound = (int)FastMath.sqrt(n);
            boundedTrialDivision(n, bound, factors);
            return factors;
        }

        /// <summary>
        /// Miller-Rabin probabilistic primality test for int type, used in such a way that a result is always guaranteed.
        /// <para>
        /// It uses the prime numbers as successive base therefore it is guaranteed to be always correct.
        /// (see Handbook of applied cryptography by Menezes, table 4.1)
        /// </summary>
        /// <param name="n">number to test: an odd integer &ge; 3</param>
        /// <returns>true if n is prime. false if n is definitely composite.</returns>
        public static Boolean millerRabinPrimeTest(int n)
        {
            int nMinus1 = n - 1;
            int mask = 1;
            int s = 32;
            for (int i = 0; i < 32; ++i, mask <<= 1) //number of trailing zeros
            {
                if ((n & mask) != 0)
                {
                    s = i;
                }
            }
            int r = nMinus1 >> s;
            //r must be odd, it is not checked here
            int t = 1;
            if (n >= 2047)
            {
                t = 2;
            }
            if (n >= 1373653)
            {
                t = 3;
            }
            if (n >= 25326001)
            {
                t = 4;
            } // works up to 3.2 billion, int range stops at 2.7 so we are safe :-)
            BigInteger br = new BigInteger(r);
            BigInteger bn = new BigInteger(n);

            for (int i = 0; i < t; ++i)
            {
                BigInteger a = new BigInteger(SmallPrimes.PRIMES[i]);
                BigInteger bPow = BigInteger.ModPow(a, br, bn);
                int y = (Int32)bPow;
                if ((1 != y) && (y != nMinus1))
                {
                    int j = 1;
                    while ((j <= s - 1) && (nMinus1 != y))
                    {
                        long square = ((long)y) * y;
                        y = (int)(square % n);
                        if (1 == y)
                        {
                            return false;
                        } // definitely composite
                        ++j;
                    }
                    if (nMinus1 != y)
                    {
                        return false;
                    } // definitely composite
                }
            }
            return true; // definitely prime
        }
    }
}
